﻿ The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions

The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions

by: Shing-Tung Yau and Steve Nadis Read in 2011-2012

xii Brian Greene was one of Yau's postdocs
2 He seems to overlook the possibility that extra dimensions may be large and still not be noticeable.
5 "One thing that higher dimensions lead to is greater complexity."
9 The development of the concept of spacetime
10 "In Einstein's theory, it takes ten numbers—or ten fields—to precisely describe the workings of gravity in four dimensions. The force can be represented most succcinctly by taking those ten numbers and arranging them in a four-by-four matrix more formally known as a metric tensor—a square table of numbers that serves as a higher-dimensional analogue of a ruler."
11 "If there really is a fifth dimension—an entirely new direction to move at every point in our familiar four-dimensional world—how come nobody has seen it?"
11 "The obvious explanation is that this dimension is awfully small."
11 Hose pipe analogy; flawed worse than usual
27 "A two-dimensional surface like this, according to Gauss, has two principal curvatures running in directions that are orthogonal to each other"
30 Metric tensor and Reimann metric
56 "the positive mass conjecture, or positive energy conjecture...states that in any isolated physical system, the total mass or energy must be positive."
56 "spacetime cannot be stable unless its overall mass is positive."
59 "gravity can interact with itself and, in the process, create mass"
59 "In general relativity, mass can only be defined globally. In other words, we think in terms of the mass of an entire system, enclosed in a figurative box, as measured from far, far away (from infinity, actually). In the case of "local"mass—the mass of a given body, for instance—there is no clear definition yet, even though this may seem like a simpler issue to the layperson. (Mass density is a similarly ill-defined concept in general relativity.)"
60 "the curvature of a sphere is inversely proportional to the radius squared"
68 "In a spacetime of 3 + 1 dimensions (three spatial dimensions and one of time), for instance, "electric fields and magnetic fields look similar," Donaldson says, "But in other dimensions, they are geometrically distinct object. One is a tensor [which is a kind of matrix] and the other is a vector, and you can't really compare them. Four dimensions is a special case in which both are vectors. Symmetries appear there that you don't see in other dimensions.""
77 "The question Calabi asked, though original in its details, is of the form we often ask in geometry—namely, given a general topology, or rough shape, what sort of precise geometric structures are allowed?" Good, this type of question needs to be explored
79 "Calabi...wanted to know, among other things, whether a certain kind of complex manifold—a space that was compact (or finite in extent) and "Kähler"—that satisfied specific topological conditions (concerning an intrinsic property known as a "vanishing first Chern class") could also satisfy the geometrical condition of having a Ricci-flat metric."
81 "One important feature of manifolds is their smoothness. It's built into the definition, because if every local patch on a surface looks Euclidean, then your surface has to be smooth overall."
81 "We call such a point [i.e. where things are not even locally Euclidean, such as a place where two lines intersect], a topological singularity because it can never be smoothed over. No matter how small you draw the neighborhood around that point, the cross made by the intersecting lines will always be there."
81 "a space can have an infinite number of singularities and still qualify in our eyes as a kind of manifold—what we call a singular space or singular manifold, which lies at the limit of a smooth manifold. Rather than two lines intersecting in a point, think instead of two planes intersecting in a line."
82 Good candidate for modeling in 4D
82 "The value of complex numbers for describing waves makes these numbers crucial to physics. In quantum mechanics, every particle in nature can also be described as a wave, and quantum theory itself will be an essential component of any theory of quantum gravity—any attempt to write a so-called theory of everything. For that task, the ability to describe waves with complex numbers is a considerable advantage."
83 "When a tiny loop of string—the basic unit of string theory—moves through higher-dimensional spacetime, it sweeps out a surface that is none other than a Riemann surface. Such surfaces have proven to be quite useful for doing computations in string theory, making them one of the most studied surfaces in theoretical physics today."
83 Mapping of Riemann surfaces is conformal, i.e. angle preserving
83 Riemann surfaces must be orientable, i.e. no Möbius strips
84 Checkerboard covering of 2D Riemann surfaces clearly shows conformality and conformal mapping
85 He refers to Riemann surfaces as 1D meaning complex coordinates whereas there are really two real coordinates.
85 "on a higher-dimensional complex manifold, angles are not preserved during movements between one patch of the manifold and another and between one coordinate system and another."
85 "Kähler geometry...enables us (among other things) to measure distance using complex numbers."
85 "In a flat space, where all the coordinate axes are perpendicular to each other, we can simply use the Pythagorean formula to compute distances. In curved spaces, where coordinate axes are not necessarily perpendicular,...[d]istance calculations...involve metric coefficients—a set of numbers that vary from point to point in space and also depend on how the coordinate axes are oriented. Selecting one orientation for the coordinate axes rather than another will change those numbers."" This is the metric tensor
86 "there are sixteen numbers in all because the metric tensor in this case is a four-by-four matrix. However, the metric tensor is always symmetrical around a diagonal axis running from the upper left corner of the matrix to the lower right. There are four numbers on the diagonal itself, and two sets of six numbers on either side of the diagonal that are the same. owing to this symmetry, we often need to concern ourselves with just ten numbers—four along the diagonal and six on either side—rather than all sixteen.)"
88 "Kähler manifolds are a subclass of a set of complex manifolds known as Hermitian manifolds. On a Hermitian manifold, you can put the origin of a complex coordinate system at any point, such that the metric will look like a standard Euclidean metric at that point. But as you move away from that point, the metric becomes increasingly non-Euclidean"
89 "the only compact manifolds that are totally flat happen to be donuts or tori or close relatives thereof, and this holds for any dimension of two or higher."
90 The "J" operation: multiplying a complex vector by i.
91 "On a Kähler manifold, the J operation remains invariant under parallel transport. That's not the case for complex manifolds in general. Another way of putting it is that on a Kähler manifold, parallel-transporting a vector and then transforming it by the J operation is the same as transforming the original vector by J and then parallel-transporting it. These two operations commute"
94 "The Gauss-Bonnet formula holds that the total Gauss curvature of such surfaces equals 2π times the Euler characteristic of that surface. The Euler characteristic—known as X ("chi") –in turn, is equal to 2-2g where g is the genus or number of "holes" or "handles" that a surface has. The Euler characteristic of s two-dimensional sphere, for example, which has zero holes, is 2. Euler had previously devised a separate formula for finding the Euler characteristic of any polyhedron: X=V-E+F, where V is the number of vertices, E the number of edges, and F the number of faces."
98 Ricci curvature and sectional curvature
99 "The sectional curvature completely determines the Riemann curvature, which in turn encodes all the curvature information you could possibly want about a surface. In four dimensions, this takes twenty numbers (and more for higher dimensions). The Riemann curvature tensor can itself be split into two terms, the Ricci tensor and something called the Weyl tensor,...[O]f the twenty numbers or components needed to describe four-dimensional Riemann curvature, ten describe Ricci curvature while ten describe Weyl curvature."
99 "The Ricci curvature tensor, a key term in the famous Einstein equation, shows how matter and energy affect the geometry of spacetime. In fact, the left hand side of this equation consists of what's called the modified Ricci tensor, whereas the right-hand side of the equation consists of the stress energy tensor, which describes the density and flow of matter in spacetime."
99 The Calabi Conjecture states that there is a non-trivial solution to the Vacuum Einstein Equation.
119 "I had confirmed the existence of many fantastic, multidimensional shapes (now called Calabi-Yau spaces) that satisfy the Einstein equation in the case where matter is absent. I had produced not just a solution to the Einstein equation, but also the largest class of solutions to that equation that we know of."
123 "It turns out that once you know a metric exists, there are all kinds of consequences. You can use that knowledge to work backward and deduce things about manifold topology, without knowing the exact metric. We can use those properties, in turn, to identify unique features of a manifold..."
123 "The Calabi-Yau equations not only satisfy the Einstein equations, they do so in a particularly elegant way..."
125 "[String] theory tells you exactly how many dimensions are needed for the job, and that number is ten—the four dimensions of the "conventional" spacetime we probe with our telescopes, plus six extra dimensions."
125 "[Nine] spatial dimensions and one of time [is] where the mathematical requirements of string theory are satisfied."
126 "Part of the challenge is to see whether string theory can explain why the universe looks the way it does. That explanation must account for the fact that we inhabit a spacetime that looks four-dimensional, while the theory insists it's actually ten-dimensional. The answer to that apparent discrepancy, according to string theory, lies in compactification."
126 "...a cylinder is the product of a line and a circle. Moreover, the cylinder, as we've seen, is how the Kaluza-Klein concept is often illustrated. If you start by depicting our four-dimensional spacetime as an infinite line that stretches forever in both directions and then snip the line and magnify one of the ends, you'll see that the line actually has some breadth and is more accurately described as a cylinder, albeit one of minuscule radius. And it is within this circle of tiny radius that the fifth dimension of Kaluza-Klein theory is hidden. String theory takes that idea several steps further, arguing in effect that when you look at the cross-section of this slender cylinder with an even more powerful microscope, you'll see six dimensions lurking inside instead of just one." Wait a minute! Who exactly is viewing this tube from afar seeing it as a line? Who is looking through the microscope? And in how many dimensions does each of them reside in? This well-worn "hose pipe" analogy fails to explain anything.
128 K3 surfaces.
128 "Andrew Strominger...[and] Philip Candelas...knew that the internal space of those shapes had to be compact (to get down from ten dimensions to four) and that the curvature had to satisfy both the Einstein gravity equations and the symmetry requirements of string theory." Saying "they knew" is pretty strong language.
129 Holonomy
131 "A Calabi-Yau manifold...belongs to the much more restrictive SU(n) holonomy group, which stands for the special unitary group of n complex dimensions."
131 "if you want to satisfy the Einstein equations as well as the supersymmetry equations—and if you want to keep the extra dimensions hidden, while preserving super-symmetry in the observable world—Calabi-Yau manifolds are the unique solution." What if you drop the hidden dimension restriction?
144 "A hypersurface is a submanifold—that is, a surface of one dimension lower than the ambient space in which it sits"
148 Branes
148 "Strominger agrees that the "notion of dimension is not absolute." He compares string theory and M-theory to different phases of water." Rather than phases, a better explanation might be the existence of several layers of embedded manifolds, or submanifolds, just as we have examples of both lines and planes here in our 3D space.
149 "Researchers have tried to go from eleven dimensions directly to four by compactifying on a seven-dimensional...manifold"
150 "eleven-dimensional spacetime is treated as the product of ten-dimensional spacetime and a one-dimensional circle. We compactify the circle, shrinking it down to a minuscule radius, which leaves us with ten dimensions. We then take those ten dimensions and compactify on a Calabi-Yau manifold, as usual, to get down to the four dimensions of our world." I think this approach is totally unnecessary. Simply considering submanifolds will do the job.
150 "This approach—pioneered by Witten, Horava, Burt Ovrut, and others—is called heterotic M-theory. It has been influential in introducing the concept of brane universes, in which our observable universe is thought to live on a brane" That's the idea. Now just generalize it and you needn't be limited to Calabi-Yau manifolds.
150 "So for now, at least, it appears that all roads lead through Calabi-Yau." But I don't think they need to be small.
161 "In general, the Betti number tells you the number of ways you can cut all the way around a two-dimensional surface without dividing it into two pieces."
165 "Physicists have a way of associating a quantum field theory with a given manifold."
165 Quantum cohomology
165 Type IIA and IIB variants of M-Theory or string theory.
166 "around the year 200 B.C. by the Greek mathematician Apollonius...asked how many circles can be drawn that are tangent to three given circles."
172 "Mathematics—despite the popular image—is not a strictly intellectual pursuit, done in total isolation, divorced from politics, ambition, competition, and emotion."
173 "The SYZ work attempts to provide a geometric interpretation of mirror symmetry, whereas homological mirror symmetry takes a more algebraic approach."
174 "If SYZ is correct...a Calabi-Yau can essentially be divided into two three-dimensional spaces that are highly entangled. One of these spacesis a three-dimensional torus. If you separate the torus from the other part, "invert" it (by switching its radius from r to 1/r), and reassemble the pieces, you'll have the mirror manifold of the original Calabi-Yau."
174 "one can also think of those D-branes as being the submanifolds rather than just wrapping them. Physicists tend to think in terms of branes, whereas mathematicians are more comfortable with their own terminology."
175 "special Lagrangian submanifolds...have half the dimension of the space within which they sit, and they have the additional attribute of being length-, area-, or volume-minimizing, among other properties."
175 "the simplest possible Calabi-Yau space [is] a two-dimensional torus or donut. The special Lagrangian submanifold in this case will be a one-dimensional space or object consisting of a loop through the hole of hte donut. And since that loop must have the minimum length, it must be a circle, the smallest possible circle going through that hole"
175 Parameterizing special Lagrangian submanifolds
175 "If we go up by one complex dimension from two real dimensions to four, the Calabi-Yau becomes a K3 surface. Instead of being circles, the submanifolds in this case are two-dimensional tori"
176 Good diagrams showing the SYZ conjecture
177 T duality, winding numbers, quantized momentum and winding
177 "T in T duality stands for tori."
177 "we titled our original SYZ paper "Mirror Symmetry Is T Duality."
179 "SYZ is still a conjecture that has only been proved in a few select cases but not in a general way."
180 Symplectic geometry and algebraic geometry
186 "If the radiation given off by the black hole is strictly thermal and is thus lacking in information content, then the information originally stored within a black hole...will disappear when the black hole evaporates. That would violate a fundamental tenet of quantum theory, which holds that the information of a system is always preserved."
186 Determinism, black holes, and entropy
188 Boltzmann, statistical mechanics and thermodynamics
188 Boltzmann's definition of entropy
193 "Although branes of different dimensions look different from each other, none of them have subcomponents or can be broken down further. In the same way, string theory holds that the string (a one-dimensional brane in M-theory parlance) is all there is and cannot be subdivided into smaller pieces."
203 "The Standard Model weaves together three forces and their associated symmetry (or gauge) groups:special unitary group 3, or SU(3), which corresponds to the strong force; special unitary group 2, or SU(2), which corresponds to the weak force; and the first unitary group, or U(1), which corresponds to the electromagnetic force.:
203 "U(1)...involves all the rotations you can do to a circle that is sitting in a two-dimensional plane."
203 "SU(2) relates to rotations in three dimensions, and SU(3), which is more abstract, very roughly involves rotations in eight dimensions. (The rule of thumb here is that any group SU[n] has a symmetry of dimension n²-1.) The dimensions of the three subgroups are additive, which means that the overall symmetry of the Standard Model is twelve-dimensional (1 + 3 + 8 = 12)."
203 "As solutions to the Einstein equation, Calabi-Yau manifolds of a particular geometry can help us account for the gravity part of our model."
204 Gauge Theories
206 "...a two-dimensional sphere or globe...has rotational symmetry in three dimensions and belongs to the symmetry group SO(3). (The term SO here is short for special orthogonal group, becuase it describes rotations around an orthogonal axis.) You can take that sphere and spin it around any of three axes—x, y, and z—and it will still look the same."
206 "...we can break that symmetry in three dimensions by insisting that one point must always stay fixed...In this way, the threefold symmetry of the sphere has been broken and reduced to a one-dimensional symmetry, U(1)."
207 "The other fields, which correspond to the broken symmetries, don't disappear entirely. By virtue of being turned on, they'll reside at a high-energy regime that puts them far beyond our reach, totally inaccessible to us. You might say the extra symmetries of E8 are hidden away in the Calabi-Yau." Then again, you might not. Fixing the North Pole doesn't mean that the earth is "compactified" down to microscopic size.
207 "Bundles are defined as groups of vectors attached to every point on the manifold. The simplest type of bundle is known as the tangent bundle." Do you really mean groups? Or do you just mean bunches.
207 "...the tangent bundle of a two-dimensional sphere is actually a four-dimensional space, because the tangent space has two degrees of freedom"
214 "...we call two curves or cycles homologous if they are of the same dimension and bound a surface or manifold of one dimension higher."
216 "The force particles are...literally part of the force fields, and the number of symmetry dimensions in each gauge field corresponds to the number of particles that communicate the force. Thus the strong force, which is endowed with eight-dimensional SU(3) symmetry, is mediated by eight gluons; the field of the weak force, which is endowed with three-dimensional SU(2) symmetry, is mediated by three particles, the W+, W-, and Z bosons; and the electromagnetic field, which is endowed with one-dimensional U(1) symmetry, is mediated by a single particle, the photon."
217 "...Vincent Bouchard of the University of Alberta and Donagi, as well as Ovrut and his colleagues, have produced models that appear to get a lot of things right. Both groups claim to get the right gauge symmetry group, the right supersymmetry, chiral fermions, and the right particle spectrum—three generations of quarks and leptons, plus a single Higgs particle, and no exotic particles, such as extra quarks or leptons that are not in the Standard Model."
219 "[To calculate] the Yukawa coupling...You chop the manifold up into tiny patches and determine the value of the function at each patch. Then you add up all those values and divide by the number of patches, and you'll get the average. But while that approach can carry you pretty far, it won't give you exactly the right answer. The problem is that the space we're working on here, the Calabi-Yau manifold, is really curvy...:" The problem might go away if you considered astronomically large Calabi-Yau manifolds.
219 "...we need the metric, which tells us the manifold's geometry in exacting detail. The only catch here, as we've said many times before, is that no one has yet figured out a way to calculate the Calabi-Yau metric explicitly, which is to say, exactly." Here again, considering large Calabi-Yau manifolds might simplify the problem, maybe even reducing it to the Euclidean metric.
220 "When you put something like a manifold in a bigger space, the subspace automatically inherits a metric (what we call an induced metric) from the background space."
221 "In the 1950s, John Nash had proved that if you put a Riemannian manifold in a space of high enough dimensions, you can get any induced metric that you want. The Nash embedding theorem...only applies to real manifolds sitting in real space. In general, the complex version of Nash's theorem is not true. But I suggested that a complex version of the theorem might be true under certain circumstances. I argued...that...the Ricci-flat metric can always be induced, and can always be approximated, by embedding the manifold in a background or projective space of sufficiently high dimension."
222 The caption on figure 9.6 seems goofy to me. If the square is considered 1D, then by bending the lines to form a square, it already has to be embedded into the 2D space of the square. Stretching it over the sphere then embeds it into the 3D space of the sphere, both the 2D square and its 1D perimeter
227 "The size and the shape of any manifold with holes in it are determined by parameters called moduli. A two-dimensional torus, for example, is in many ways defined by two independent loops or cycles, one going around the hole and another going through. The moduli, by definition, measure the size of the cycles, which themselves govern both the size and shape of the manifold."
231 "The paper in question (dubbed KKLT after its authors, Shamit Kachru, Renata Kallosh, and Andrei Linde—all from Stanford—and Sandip Trivedi of the Tata Institute in India) is generally considered the first publication to show a consistent way of stabilizing all the moduli of the Calabi-Yau, both the shape moduli and size moduli."
231 "If the small, invisible dimensions suddenly sprang free and expanded, we'd then be living in a spacetime of ten large dimensions, with ten independent directions to move in or to search for our missing keys, and we know our world doesn't look like that."
232 "When given a chance, all fields will try to spread out and get dilute."
232 "In the KKLT scenario, branes provided a possible mechanism for realizing the universe we see—a universe that's influenced by inflation to a large degree."
232 "Another milestone achieved by KKLT was providing a string theory description of how our universe might be endowed with a positive vacuum energy—sometimes called dark energy"
233 "If you take a Calabi-Yau manifold and throw in D-branes and fluxes, you may have all the ingredients you need in principle to get the Standard Model, inflation, dark energy, and other things we need to explain our world."
234 "...because Calabi-Yau manifolds are, by definition, solutions to the vacuum Einstein equations, each of those solutions, which involve different ways of incorporating fluxes and branes, corresponds to a universe with a different vacuum state and, hence, a different vacuum energy. Now here's the kicker: A fair number of theorists believe that all these possible universes might actually exist."
235 "...it is thought that the bulk of the vacuum energy is used to keep the six extra dimensions of string theory curled up in such spaces rather than allowing them to expand to infinity."
244 "String theorists introduced fluxes to get rid of massless scalar fields and thereby stabilize the size and shape of a Calabi-Yau manifold."
244 "From the standpoint of string theory, one of the chief roles intended for these manifolds, be they Calabi-Yau or non-Kähler, is compactification—reducing the theory's ten dimensions to the four of our world. The easiest way to partition the space is to cut it cleanly, splitting it into four-dimensional and six-dimensional components. That's essentially the Calabi-Yau approach. We tend to think of these two components as wholly separate and noninteracting. Ten-dimensional spacetime is thus the Cartesian product of its four- and six-dimensional parts, and as we've seen, you can visualize it with the Kaluza-Klein-style model we discussed in Chapter 1: In this picture, our infinite, four-dimensional spacetime is like an infinitely long line, except that this line has some thickness—a tiny circle wherein the extra six dimensions reside. So what we really have is the Cartesian product of a circle and a line—a cylinder, in other words."
248 "...the famous Einstein equation that encapsulates the theory of general relativity is really a set of ten field equations that describe gravity as the curvature of spacetime caused by the presence of matter and energy, even though it can be written as a single tensor equation."
258 "In the most well-developed models we have today, the energy of the vacuum is a consequence of the compactification of the extra dimensions. Put in other terms, the dark energy we've heard so much about isn't just driving the cosmos apart in some kind of madcap accelerative binge: Some, if not all, of that energy goes into keeping the extra dimensions wound up tighter than the springs of a Swiss watch, although in our universe, unlike in a Rolex, this is done with fluxes and branes."
258 "...compactification...as we've discussed, is one of the biggest challenges in string theory: If the theory depends on our universe's having ten dimensions, how come we only see four? String theorists have been hard-pressed to explain how the theory's extra dimensions are so well concealed, because, as Linde has noted, all other things being equal, the dimensions would rather be big."
263 de Sitter space, entropy, cosmological constant, and inverse square laws
266 "It is not surprising that few investigators are strongly inclined to carry this [subject of statistical mechanics, entropy, and the fate of the universe] much further, since we're talking about highly speculative events in model-dependent scenarios that are not readily testable and are expected to happen on a timescale just shy of forever. That's hardly the ideal prescription for getting grant money or, for younger researchers, gaining the admiration of their elders and, more importantly, securing tenure." It seems that the motivations of scientists are less than ideal.
267 "With ten spacetime dimensions to play with, and six new directions in which to roam, life would have possibilities we can't even fathom."
286 "Gravity as we know it in four-dimensional spacetime obeys an inverse square law. The gravitational influence of a body drops off with the square of the distance from it. But if we added another dimension, gravity would drop off as the cube of the distance." I don't think this is true.
301 "Seiberg-Witten equations...categorize and classify all possible shapes." in 4D. See also page 68
304 "Quantum electrodynamics is an example of an extremely successful theory that is not mathematically well defined." – Max Tegmark
304 "Cliff Taubes...: "Physics is the study of the world, while mathematics is the study of all possible worlds."
306 "I get to think about what's possible---not only 'all possible worlds,' as Cliff put it, but the even broader category of all possible spaces." I suggest you think about large nearly flat hyperspaces
308 The need for Quantum Geometry
308 "...the problem has more to do with physics than with math. For one thing, the Planck scale where all this trouble starts is not a mathematical concept at all." It would be in my Practical Number System
308 "...the fact that classical geometry breaks down at the Planck scale doesn't mean there's anything wrong with the math per se." I think it does. What's wrong is the inclusion of the Axiom of Choice or any other admission of infinity.
308 "The techniques of differential calculus that underlie Riemannian geometry, which in turn provides the basis for general relativity, do not suddenly stop working at a critical length scale. Differential geometry is designed by its very premise to operate on infinitesimally small lengths that can get as close to zero as you want." That would change for the better in my Practical Number System
309 "David Morrison, UC Santa Barbara." Contact him
312 "Geometry should be ranked not with arithmetic, which is purely a priori, but with mechanics"
313 "Classical Riemannian geometry is not capable of describing physics at the quantum level. Instead we seek a new geometry, a more general description, that applies equally well to a Rubik's Cube as it does o a Planck-length string. The question is how to proceed." I suggest we pursue the project of Practical Numbers.